Spontaneous magnetization and structure formation in a spin-1 ferromagnetic Bose-Einstein condensate

Motivated by recent experiments involving the nondestructive imaging of magnetization of a spin-1 ${}^{87}\mathrm{Rb}$ Bose gas [J. M. Higbie et al., Phys. Rev. Lett., 95, 050401 (2005)], we address the question of how the spontaneous magnetization of a ferromagnetic Bose-Einstein condensate occurs in a spin-conserving system. Due to competition between the ferromagnetic interaction and the total spin conservation, various spin structures such as staggered magnetic domains and helical and concentric ring structures are formed, depending on the geometry of the trapping potential.

Figure 1

(a) (Color) Time evolution of the relative population of the $m=0$ component, $\int d\mathbf{r}{\mid {\psi }_0\mid }^2∕N$, and (b) the column density distribution of the total density, $\int dy{\sum }_{m=-1}^1{\mid {\psi }_m\mid }^2$, and those of the $m=1$, 0, and $-1$ components, $\int dy{\mid {\psi }_m\mid }^2$, from top to bottom. The trap frequencies are ${\omega }_{\perp }=2\pi \times 245\mathrm{Hz}$ and ${\omega }_z=2\pi \times 4\mathrm{Hz}$. The magnetic field of $54\mathrm{mG}$ is applied in the $z$ direction. The initial state is given by ${\psi }_1=0$, ${\psi }_0=\sqrt{1-{10}^{-4}}{\psi }_{\mathrm{g}}$, and ${\psi }_{-1}=0.01{\psi }_{\mathrm{g}}$, where ${\psi }_{\mathrm{g}}$ is the $m=-1$ ground state. The size of each strip is $600\times 16$ in units of ${(\hslash ∕m{\omega }_{\perp })}^{1∕2}\simeq 0.69\mu \mathrm{m}$.

Figure 2

(a) (Color) Time evolution of the three components of the total spin per atom, ${\mathit{F}}^{\mathrm{tot}}∕N\equiv \int d\mathbf{r}\mathit{F}∕N$, and (b) the radial and axial distributions of the three components of the mean spin vector, $\mathit{F}∕n$. The conditions are the same as those given in Fig. 1, and the data are taken on the rotating frame of reference, in which the $38\text{−}\mathrm{kHz}$ Larmor precession in the $x\text{−}y$ plane is eliminated for the sake of clarity. Each strip in (b) indicates the radial (ordinate) and axial (abscissa) distribution, where the bottom left corner is the origin $(r=z=0)$. From $t=440$ to 580, only the $F_x∕n$ distributions are shown at every ${\omega }_{\perp }\Delta t=10$, because the other components are negligibly small. After ${\omega }_{\perp }t=600$, $F_x∕n$, $F_y∕n$, and $F_z∕n$ are shown from top to bottom. The size of each strip is $300\times 6$ in units of ${(\hslash ∕m{\omega }_{\perp })}^{1∕2}$.

Figure 3

(Color) The mean spin vectors $\mathit{F}∕n$ at $r=0$ seen from the $-y$ direction, where the vertical axis is the $z$ axis. The conditions are the same as those given in Fig. 1. The length of the spin vector is proportional to $\mid \mathit{F}\mid ∕n$, and the color represents $F_x∕n$ according to the gauge shown at the top left corner. The spin vector is displayed from $z=0\text{to}z=54\mu \mathrm{m}$ in (a)–(c) and from $z=35\mu \mathrm{m}\text{to}z=89\mu \mathrm{m}$ in (d). The Larmor precession in the $x\text{−}y$ plane is eliminated for clarity of presentation.

Figure 4

(Color) The total density and the three components of the mean spin vector $\mathit{F}∕n$ of the two-dimensional system (from left to right panel), where the abscissa and ordinate refer to the $x$ and $y$ coordinates in real space. The size of each image is $48\times 48$ in units of ${(\hslash ∕m{\omega }_{\perp })}^{1∕2}$. The color for the mean spin vector refers to the gauge. The bottom panel illustrates the direction of the spin at ${\omega }_{\perp }t=26$, where the color represents $F_x∕n$ as shown in Fig. 3. The Larmor precession in the $x\text{−}y$ plane is eliminated.

Figure 5

(Color) The three components of the mean spin vector $F_x∕n$, $F_y∕n$, and $F_z∕n$ [from top to bottom panel in (a) and from left to right panel in (b)] for the initial state ${\psi }_1=\sqrt{1+0.01}{\psi }_{\mathrm{g}}$, ${\psi }_0=0$, and ${\psi }_{-1}=\sqrt{1-0.01}{\psi }_{\mathrm{g}}$, where ${\psi }_{\mathrm{g}}$ is the $m=-1$ ground state. The other conditions are the same as those in Fig. 2 for (a) and in Fig. 4 for (b).